Let d 2 D = {1, . . . , 9}, and let k be a positive integer with gcd(k, 10d) = 1. Define a sequence {sn(k, d)}1 n=1 by sn(k, d) := k dd . . . d | {z } n k. We say k is a d-composite sandwich number if sn(k, d) is composite for all n 1. For a d-composite sandwich number k, we say k is trivial if sn(k, d) is divisible by the same prime for all n 1, and nontrivial otherwise. In this paper, we develop a simple criterion to determine when a d-composite sandwich number is nontrivial, and we use it to establish many results concerning which types of integers can be d-composite sandwich numbers. For example, we prove that there exist infinitely many primes that are simultaneously trivial d-composite sandwich numbers for all d 2 D. We also show that there exist infinitely many positive integers that are simultaneously nontrivial d-composite sandwich numbers for all d 2 D, where D ? D with |D| = 4 and D 6= {3, 6, 7, 9}.
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机译：让D 2 D = {1，。 。 。 ，9}，让K是带GCD（k，10d）= 1.通过sn（k，d）：= k dd的序列{sn（k，d）} 1 n = 1的正整数。 。 。 D | {z} n k。我们说K是一个D-复合夹层编号，如果SN（k，d）是所有n的复合材料。对于D-复合夹心编号k，我们说k是微小的，如果sn（k，d）通过相同的方式所有n 1的素数，否则是非。在本文中，我们开发了一种简单的标准，以确定D-Composite Sandwich编号是否是非虚拟性的，并且我们使用它来建立许多关于哪种类型的整数可以是D-Composite Sandwich数字的结果。例如，我们证明，对于所有D 2 D同时存在无限的许多素数，这也是同时进行D-Composite Sandwich编号的多重型D-Composite Sandwich编号。我们还表明，存在多种正整数，这些正整数是所有D 2 D的非竞争D-复合夹层编号的阳性整数。 ，其中d？ d | D | = 4和D 6 = {3,6,7,9}。
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